ascendModule -- find the smallest submodule of free module containing a given submodule which is compatible with a given Cartier linear map

Synopsis

• Usage:

  V = ascendModule(e, M, U)

  B = ascendModule(e, A, U)

• Inputs:
  • M, a module, a submodule of a free module R^n, where R is a polynomial ring over a finite field of characteristic p
  • A, a matrix, an {n\times m} matrix with entries in R
  • U, a matrix, an {n\times n} matrix with entries in R
  • e, an integer,
• Outputs:
  • V, a module, the smallest submodule of R^n containing M such that U^{1+p+\ldots+p^{e-1}} V \subseteq V^{[p^e]}
  • B, a matrix, whose image is the smallest submodule V of R^n containing the image of A such that U^{1+p+\ldots+p^{e-1}} V \subseteq V^{[p^e]}

Description

i1 : R = ZZ/2[a,b,c,d];

i2 : A = matrix {{b*c, a, 0}, {a^2* d, d^2 , c + d}};

             2      3
o2 : Matrix R  <-- R

i3 : M = image A;

i4 : U = matrix {{a^4  + a*b*c^2  + a*b*c*d, a^2* b}, {a^2*c*d^3 , a^3* c*d + a^3 *d^2  + b*c*d^3}};

             2      2
o4 : Matrix R  <-- R

i5 : ascendModule(1, M, U)

o5 = image | 0   a  bc  |
           | c+d d2 a2d |

                             2
o5 : R-module, submodule of R

i6 : ascendModule(1, A, U)

o6 = | 0   a  bc  |
     | c+d d2 a2d |

             2      3
o6 : Matrix R  <-- R